## 3.9 Concluding Discussion

Integral equation methods such as the boundary element method
are becoming increasingly popular as methods
for the numerical solution of linear elliptic partial differential
equations such as the Laplace equation. The application of
(discrete) collocation to the integral equation formulation
requires the computation of the discrete operators.
Fortran subroutines for the evaluation of the discrete
Laplace integral operators resulting from the use of constant
elements and the most simple boundary approximation to two-dimensional,
three-dimensional and axisymmetric problems have been described and
demonstrated.
The computational cost of a subroutine call is roughly proportional to
the number of points in the quadrature rule. Ideally, the quadrature rule
should have as few points as possible but must still
give a sufficiently accurate approximation to the discrete operator.
The efficiency of the overall method will generally be enhanced by
varying the quadrature rule with the distance between the
point **p** and the element, the size of the element and
the wavenumber.
The use of the subroutines
would be improved by a method for the automatic selection of
the quadrature rule so that the discrete operator can be computed with the
minimum number of quadrature points for some predetermined accuracy.
The subroutines have been designed to be easy-to-use, flexible, reliable
and efficient. It is the intention that the subroutines are to be
used as a `black box' which can be utilised either for further analysis
of integral equation methods or in software for the solution
of practical physical problems which are governed by the
Laplace equation.

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